Here's some handy advice: Don't use any type of cleaner on the bass strings!!!! Take it from someone who turned copper strings into beautiful, like-new beauty and at the same time ruined them. Anything you do that causes ANYTHING to work between the coils will kill the string (including steel wool, which will cause the fine pulverized powdered verdigris (if that's what it's called) to settle between the coils. Once this happens, you're in big trouble. My advice with copper wound strings -- leave them alone. If they're dull and tarnished, replace the light bulb in your room with lower wattage.
The steel strings can be cleaned. There is a special cleaner that is like an ink eraser with a handle. Works great, and is made for the job. Steel wool is OK, but take it easy. Steel against steel can scratch. It's best to use something softer than steel, so that only the gunk (tarnish / oxidation) is removed, with no possibility of any effect on the steel.
I'm sure there are other opinions on this, but my opinion is that tarnished strings are not broken -- and one should not try to fix something not broke.